# 1911 Encyclopædia Britannica/Capillary Action

 Capillary Action
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CAPILLARY ACTION.[1] A tube, the bore of which is so small that it will only admit a hair (Lat. capilla), is called a capillary tube. When such a tube of glass, open at both ends, is placed vertically with its lower end immersed in water, the water is observed to rise in the tube, and to stand withing the tube at a higher level than the water outside. The action between the capillary tube and the water has been called capillary action, and the name has been extended to many other phenomena which have been found to depend on properties of liquids and solids similar to those which cause water to rise in capillary tubes.

The forces which are concerned in these phenomena are those which act between neighbouring parts of the same substance, and which are called forces of cohesion, and those which ac between portions of matter of different kinds, which are called forces of adhesion. These forces are quite insensible between two portions of matter separated by any distance which we can directly measure. It is only when the distance becomes exceedingly small that these forces become perceptible. G.H. Quincke (Pogg. Ann. cxxxvii. p. 402) made experiments to determine the greatest distance at which the effect of these forces is sensible, and he found for various substances distances about the twenty-thousandth part of a millimetre.

Historical.—According to J.C. Poggendorff (Pogg. Ann. ci. p. 551), Leondardo da Vinci must be considered as the discoverer of capillary phenomena, but the first accurate observations of the capillary action of tubes and glass plates were made by Francis Hawksbee (Physico-Mechanical Experiments, London, 1709, pp. 139-169; and Phil. Trans., 1711 and 1712), who ascribed the action to an attraction between the glass and the liquid. He observed that the effect was the same in thick tubes as in thin, and concluded that only those particles of the glass which are very near the surface have any influence on the phenomenon. Dr James Juri (Phil. Trans., 1718, p. 739, and 1719, p. 1083) showed that the height at which the liquid is suspended depends on the section of the tube at the surface of the liquid, and is independent of the form of the lower part of the tube. He considered that the suspension of the liquid is due to "the attraction of the periphery or section of the surface of the tube to which the upper surface of the water is contiguous and coheres." From this he showed that the rise of the liquid in tubes of the same substance is inversely proportional to their radii. Sir Isaac Newton devoted the 31st query in the last edition of his Opticks to molecular forces, and instanced several examples of the cohesion of liquids, such as the suspension of mercury in a barometer tube at more than double the height at which is usually stands. This arises from its adhesion to the tube, and the upper part of the mercury sustains a considerable tension, or negative pressure, without the separation of its parts. He considered the capillary phenomena to be of the same kind, but his explanation is not sufficiently explicit with respect to the nature and the limits of the action of the attractive force.

It is to be observed that, while these early speculators ascribe the phenomena to attraction, they do not distinctly assert that this attraction is sensible only at insensible distances, and that for all distances which we can directly measure the force is altogether insensible. The idea of such forces, however, had been distinctly formed by Newton, who gave the first example of the calculation of the effect of such forces in his theorem on the alteration of the path of a light-corpuscle when it enters or leaves a dense body.

Alex Claude Clairault (Théorie de la figure de la terre, Paris, 1808, pp. 105, 128) appears to have been the first to show the necessity of taking account of the attraction between the parts of the fluid itself in order to explain the phenomena. He did not, however, recognize the fact that the distance at which the attraction if sensible is not only small but altogether insensible. J. A. con Segner ( Comment. Soc. Reg. Götting. i. (1751) p. 301) introduced the very important idea of the surface-tension of liquids, which he ascribed to attractive forces, the sphere of whose action is so small "ut nullo adhuc sensu percipi potuerit." In attempting to calculate the effect of this surface-tension in determining the form of a drop of the liquid, Segner took account of the curvature of a meridian section of the drop, but neglected the effect of the curvature in a plane at right angles to this section.

The idea of surface-tension introduced by Segner had a most important effect on the subsequent development of the theory. We may regard it as a physical fact established by experiment in the same way as the laws of the elasticity of solid bodies. We may investigate the forces which act between finite portions of a liquid in the same way as we investigate the forces which act between finite portions of a solid. The experiments on solids lead to certain laws of elasticity expressed in terms of coefficients, the values of which can be determined only by experiments on each particular substance. Various attempts have also been made to deduce these laws from particular hypotheses as to the action between the molecules of the elastic substance. We may therefore regard the theory of elasticity as consisting of two parts. The first part establishes the laws of the elasticity of a finite portion of the solid subjected to a homogeneous strain, and deduces from these laws the equations of the equilibrium, and motion of a body subjected to any forces and displacements. The second part endeavours to deduce the facts of the elasticity of a finite portion of the substance from hypotheses as to the motion of its constituent molecules and the forces acting between them. In like manner we may by experiment ascertain the general fact that the surface of a liquid is in a state of tension similar to that of a membrane stretched equally in all directions, and prove that this tension depends only on the nature and temperature of the liquid and not on its form, and from this as a secondary physical principle we may deduce all the phenomena of capillary action. This is one step of the investigation. The next step is to deduce this surface-tension from a hypothesis as to the molecular constitution of the liquid and of the bodies that surround it. The scientific importance of this step is to be measured by the degree of insight which it affords or promises into the molecular constitution of real bodies by the suggestion of experiments by which we may discriminate between rival molecular theories.

In 1756 J.G. Leidenfrost (De aquae communis nonnullis qualitatibus tractatus, Duisburg) showed that a soap-bubble tends to contract, so that if the tube with which it was blown is left open the bubble will diminish in size and will expel through the tube the air which it contains. He attributed this force, however, not to any general property of the surfaces of liquids, but to the fatty part of the soap which he supposed to separate itself from the other constituents of the solution, and to form a thin skin on the outer face of the bubble.

In 1787 Gaspard Monge (Mémoires de l’Acad. des Sciences, 1787, p. 506) asserted that “by supposing the adherence of the particles of a fluid to have a sensible effect only at the surface itself and in the direction of the surface it would be easy to determine the curvature of the surfaces of fluids in the neighbourhood of the solid boundaries which contain them; that these surfaces would be linteariae of which the tension, constant in all directions, would be everywhere equal to the adherence of two particles, and the phenomena of capillary tubes would then present nothing which could not be determined by analysis.” He applied this principle of surface-tension to the explanation of the apparent attractions and repulsions between bodies floating on a liquid.

In 1802 John Leslie (Phil. Mag., 1802, vol. xiv. p. 193) gave the first correct explanation of the rise of a liquid in a tube by considering the effect of the attraction of the solid on the very thin stratum of the liquid in contact with it. He did not, like the earlier speculators, suppose this attraction to act in an upward direction so as to support the fluid directly. He showed that the attraction is everywhere normal to the surface of the solid. The direct effect of the attraction is to increase the pressure of the stratum of the fluid in contact with the solid, so as to make it greater than the pressure in the interior of the fluid. The result of this pressure if unopposed is to cause this stratum to spread itself over the surface of the solid as a drop of water is observed to do when placed on a clean horizontal glass plate, and this even when gravity opposes the action, as when the drop is placed on the under surface of the plate. Hence a glass tube plunged into water would become wet all over were it not that the ascending liquid film carries up a quantity of other liquid which coheres to it, so that when it has ascended to a certain height the weight of the column balances the force by which the film spreads itself over the glass. This explanation of the action of the solid is equivalent to that by which Gauss afterwards supplied the defect of the theory of Laplace, except that, not being expressed in terms of mathematical symbols, it does not indicate the mathematical relation between the attraction of individual particles and the final result. Leslie’s theory was afterwards treated according to Laplace’s mathematical methods by James Ivory in the article on capillary action, under “Fluids, Elevation of,” in the supplement to the fourth edition of the Encyclopaedia Britannica, published in 1819.

In 1804 Thomas Young (Essay on the “Cohesion of Fluids,” Phil. Trans., 1805, p. 65) founded the theory of capillary phenomena on the principle of surface-tension. He also observed the constancy of the angle of contact of a liquid surface with a solid, and showed how from these two principles to deduce the phenomena of capillary action. His essay contains the solution of a great number of cases, including most of those afterwards solved by Laplace, but his methods of demonstration, though always correct, and often extremely elegant, are sometimes rendered obscure by his scrupulous avoidance of mathematical symbols. Having applied the secondary principle of surface-tension to the various particular cases of capillary action, Young proceeded to deduce this surface-tension from ulterior principles. He supposed the particles to act on one another with two different kinds of forces, one of which, the attractive force of cohesion, extends to particles at a greater distance than those to which the repulsive force is confined. He further supposed that the attractive force is constant throughout the minute distance to which it extends, but that the repulsive force increases rapidly as the distance diminishes. He thus showed that at a curved part of the surface, a superficial particle would be urged towards the centre of curvature of the surface, and he gave reasons for concluding that this force is proportional to the sum of the curvatures of the surface in two normal planes at right angles to each other.

The subject was next taken up by Pierre Simon Laplace (Mécanique céleste, supplement to the tenth book, pub. in 1806). His results are in many respects identical with those of Young, but his methods of arriving at them are very different, being conducted entirely by mathematical calculations. The form into which he threw his investigation seems to have deterred many able physicists from the inquiry into the ulterior cause of capillary phenomena, and induced them to rest content with deriving them from the fact of surface-tension. But for those who wish to study the molecular constitution of bodies it is necessary to study the effect of forces which are sensible only at insensible distances; and Laplace has furnished us with an example of the method of this study which has never been surpassed. Laplace investigated the force acting on the fluid contained in an infinitely slender canal normal to the surface of the fluid arising from the attraction of the parts of the fluid outside the canal. He thus found for the pressure at a point in the interior of the fluid an expression of the form

$p=K + \frac{1}{2}H(1/R + 1/R'),$

where K is a constant pressure, probably very large, which, however, does not influence capillary phenomena, and therefore cannot be determined from observation of such phenomena; H is another constant on which all capillary phenomena depend; and R and R’ are the radii of curvature of any two normal sections of the surface at right angles to each other.

In the first part of our own investigation we shall adhere to the symbols used by Laplace, as we shall find that an accurate knowledge of the physical interpretation of these symbols is necessary for the further investigation of the subject. In the Supplement to the Theory of Capillary Action, Laplace deduced the equation of the surface of the fluid from the condition that the resultant force on a particle at the surface must be normal to the surface. His explanation, however, of the rise of a liquid in a tube is based on the assumption of the constancy of the angle of contact for the same solid and fluid, and of this he has nowhere given a satisfactory proof. In this supplement Laplace gave many important applications of the theory, and compared the results with the experiments of Louis Joseph Gay Lussac.

The next great step in the treatment of the subject was made by C.F. Gauss (Principia generalia Theoriae Figurae Fluidorum in statu Aequilibrii, Göttingen, 1830, or Werke, v. 29, Göttingen, 1867). The principle which he adopted is that of virtual velocities, a principle which under his hands was gradually transforming itself into what is now known as the principle of the conservation of energy. Instead of calculating the direction and magnitude of the resultant force on each particle arising from the action of neighbouring particles, he formed a single expression which is the aggregate of all the potentials arising from the mutual action between pairs of particles. This expression has been called the force-function. With its sign reversed it is now called the potential energy of the system. It consists of three parts, the first depending on the action of gravity, the second on the mutual action between the particles of the fluid, and the third on the action between the particles of the fluid and the particles of a solid or fluid in contact with it.

The condition of equilibrium is that this expression (which we may for the sake of distinctness call the potential energy) shall be a minimum. This condition when worked out gives not only the equation of the free surface in the form already established by Laplace, but the conditions of the angle of contact of this surface with the surface of a solid.

Gauss thus supplied the principal defect in the great work of Laplace. He also pointed out more distinctly the nature of the assumptions which we must make with respect to the law of action of the particles in order to be consistent with observed phenomena. He did not, however, enter into the explanation of particular phenomena, as this had been done already by Laplace, but he pointed out to physicists the advantages of the 258 method of Segner and Gay Lussac, afterwards carried out by Quincke, of measuring the dimensions of large drops of mercury on a horizontal or slightly concave surface, and those of large bubbles of air in transparent liquids resting against the under side of a horizontal plate of a substance wetted by the liquid.

In 1831 Siméon Denis Poisson published his Nouvelle Théorie de l’action capillaire. He maintained that there is a rapid variation of density near the surface of a liquid, and he gave very strong reasons, which have been only strengthened by subsequent discoveries, for believing that this is the case. He proceeded to an investigation of the equilibrium of a fluid on the hypothesis of uniform density, and arrived at the conclusion that on this hypothesis none of the observed capillary phenomena would take place, and that, therefore, Laplace’s theory, in which the density is supposed uniform, is not only insufficient but erroneous. In particular he maintained that the constant pressure K, which occurs in Laplace’s theory, and which on that theory is very large, must be in point of fact very small, but the equation of equilibrium from which he concluded this is itself defective. Laplace assumed that the liquid has uniform density, and that the attraction of its molecules extends to a finite though insensible distance. On these assumptions his results are certainly right, and are confirmed by the independent method of Gauss, so that the objections raised against them by Poisson fall to the ground. But whether the assumption of uniform density be physically correct is a very different question, and Poisson rendered good service to science in showing how to carry on the investigation on the hypothesis that the density very near the surface is different from that in the interior of the fluid.

The result, however, of Poisson’s investigation is practically equivalent to that already obtained by Laplace. In both theories the equation of the liquid surface is the same, involving a constant H, which can be determined only by experiment. The only difference is in the manner in which this quantity H depends on the law of the molecular forces and the law of density near the surface of the fluid, and as these laws are unknown to us we cannot obtain any test to discriminate between the two theories.

We have now described the principal forms of the theory of capillary action during its earlier development. In more recent times the method of Gauss has been modified so as to take account of the variation of density near the surface, and its language has been translated in terms of the modern doctrine of the conservation of energy.[2]

J.A.F. Plateau (Statique expérimentale et théorique des liquides), who made elaborate study of the phenomena of surface-tension, adopted the following method of getting rid of the effects of gravity. He formed a mixture of alcohol and water of the same density as olive oil, and then introduced a quantity of oil into the mixture. It assumes the form of a sphere under the action of surface-tension alone. He then, by means of rings of iron-wire, disks and other contrivances, altered the form of certain parts of the surface of the oil. The free portions of the surface then assume new forms depending on the equilibrium of surface-tension. In this way he produced a great many of the forms of equilibrium of a liquid under the action of surface-tension alone, and compared them with the results of mathematical investigation. He also greatly facilitated the study of liquid films by showing how to form a liquid, the films of which will last for twelve or even for twenty-four hours. The debt which science owes to Plateau is not diminished by the fact that, while investigating these beautiful phenomena, he never himself saw them, having lost his sight in about 1840.

G.L. van der Mensbrugghe (Mém. de l’Acad. Roy. de Belgique, xxxvii., 1873) devised a great number of beautiful illustrations of the phenomena of surface-tension, and showed their connexion with the experiments of Charles Tomlinson on the figures formed by oils dropped on the clean surface of water.

Athanase Dupré in his 5th, 6th and 7th Memoirs on the Mechanical Theory of Heat (Ann. de Chimie et de Physique, 1866-1868) applied the principles of thermodynamics to capillary phenomena, and the experiments of his son Paul were exceedingly ingenious and well devised, tracing the influence of surface-tension in a great number of very different circumstances, and deducing from independent methods the numerical value of the surface-tension. The experimental evidence which Dupré obtained bearing on the molecular structure of liquids must be very valuable, even if our present opinions on this subject should turn out to be erroneous.

F.H.R. Lüdtge (Pogg. Ann. cxxxix. p. 620) experimented on liquid films, and showed how a film of a liquid of high surface-tension is replaced by a film of lower surface-tension. He also experimented on the effects of the thickness of the film, and came to the conclusion that the thinner a film is, the greater is its tension. This result, however, was tested by Van der Mensbrugghe, who found that the tension is the same for the same liquid whatever be the thickness, as long as the film does not burst. [The continued coexistence of various thicknesses, as evidenced by the colours in the same film, affords an instantaneous proof of this conclusion.] The phenomena of very thin liquid films deserve the most careful study, for it is in this way that we are most likely to obtain evidence by which we may test the theories of the molecular structure of liquids.

Sir W. Thomson (afterwards Lord Kelvin) investigated the effect of the curvature of the surface of a liquid on the thermal equilibrium between the liquid and the vapour in contact with it. He also calculated the effect of surface-tension on the propagation of waves on the surface of a liquid, and determined the minimum velocity of a wave, and the velocity of the wind when it is just sufficient to disturb the surface of still water.

Theory of Capillary Action

When two different fluids are placed in contact, they may either diffuse into each other or remain separate. In some cases diffusion takes place to a limited extent, after which the resulting mixtures do not mix with each other. The same substance may be able to exist in two different states at the same temperature and pressure, as when water and its saturated vapour are contained in the same vessel. The conditions under which the thermal and mechanical equilibrium of two fluids, two mixtures, or the same substance in two physical states in contact with each other, is possible belong to thermodynamics. All that we have to observe at present is that, in the cases in which the fluids do not mix of themselves, the potential energy of the system must be greater when the fluids are mixed than when they are separate.

It is found by experiment that it is only very close to the bounding surface of a liquid that the forces arising from the mutual action of its parts have any resultant effect on one of its particles. The experiments of Quincke and others seem to show that the extreme range of the forces which produce capillary action lies between a thousandth and a twenty-thousandth part of a millimetre.

We shall use the symbol ε to denote this extreme range, beyond which the action of these forces may be regarded as insensible. If χ denotes the potential energy of unit of mass of the substance, we may treat χ as sensibly constant except within a distance ε of the bounding surface of the fluid. In the interior of the fluid it has the uniform value χ0. In like manner the density, ρ, is sensibly equal to the constant quantity ρ0, which is its value in the interior of the liquid, except within a distance ε of the bounding surface. Hence if V is the volume of a mass M of liquid bounded by a surface whose area is S, the integral

$M = \int\int\int \rho dxdydz,\;.\;.\;.\; (1)$

where the integration is to be extended throughout the volume V, may be divided into two parts by considering separately the thin shell or skin extending from the outer surface to a depth ε, within which the density and other properties of the liquid vary with the depth, and the interior portion of the liquid within which its properties are constant.

Since ε is a line of insensible magnitude compared with the dimensions of the mass of liquid and the principal radii of curvature of its surface, the volume of the shell whose surface is S and thickness ε will be Sε, and that of the interior space will be V − Sε.

If we suppose a normal ν less than ε to be drawn from the surface S into the liquid, we may divide the shell into elementary shells whose thickness is dν, in each of which the density and other properties of the liquid will be constant.

The volume of one of these shells will be Sdν. Its mass will be Sρdν. The mass of the whole shell will therefore be $S\int_0^\epsilon \epsilon _0 \rho d \nu$, and that of the interior part of the liquid (V − Sε)ρ0. We thus find for the whole mass of the liquid

$M = V_{\rho_0} - S\int_0^\epsilon (\rho_0- \rho) d \nu.\;\;.\;.\;.\; (2)$

To find the potential energy we have to integrate

$E = \int\int\int \chi\rho dxdydz.\;\;.\;.\;.\; (3)$

Substituting χρ for ρ in the process we have just gone through, we find

$M = V_{\chi_{0}\rho_0} - S\int_0^\epsilon (\chi_0\rho_0- \chi\rho) d \nu.\;\;.\;.\;.\;(4)$

Multiplying equation (2) by χ0, and subtracting it from (4),

$E - M_{\chi_0} = \chi_0 S\int_0^\epsilon (\chi- \chi_0)\rho d\nu.\;\;.\;.\;.\; (5)$

In this expression M and χ0 are both constant, so that the variation of the right-hand side of the equation is the same as that of the energy E, and expresses that part of the energy which depends on the area of the bounding surface of the liquid. We may call this the surface energy.

The symbol χ expresses the energy of unit of mass of the liquid at a depth ν within the bounding surface. When the liquid is in contact with a rare medium, such as its own vapour or any other gas, χ is greater than χ0, and the surface energy is positive. By the principle of the conservation of energy, any displacement of the liquid by which its energy is diminished will tend to take place of itself. Hence if the energy is the greater, the greater the area of the exposed surface, the liquid will tend to move in such a way as to diminish the area of the exposed surface, or, in other words, the exposed surface will tend to diminish if it can do so consistently with the other conditions. This tendency of the surface to contract itself is called the surface-tension of liquids.

### Notes

1. ^  In this revision of James Clerk Maxwell's classical article in the ninth edition of the Encyclopædia Britannica, additions are marked by square brackets.
2. ^  See Enrico Betti, Teoria della Capillarità: Nuovo Cimento (1867); a memoir by M. Stahl, “Ueber einige Punckte in der Theorie der Capillarerscheinungen,” Pogg. Ann. cxxxix. p. 239 (1870); and J.D. Van der Waal’s Over de Continuiteit van den Gasen Vloeistoftoestand. A good account of the subject from a mathematical point of view will be found in James Challis’s “Report on the Theory of Capillary Attraction,” Brit. Ass. Report, iv. p. 235 (1834).